Whenever you hear about new superweapons, check how successful their previous five tests were, and then check how realistic those tests were. Then try to understand what fundamental physical, logistical, and game-theoretic obstacles must be addressed by the weapon and under which circumstances it can actually be used.

Take the Russian land-based Topol ICBMs (operational since 1988, scheduled for decomissioning around 2021). Eighty-five test launches, 85 successes, 120 single-warhead missiles remaining. Then consider the newer Bulava missiles, which are quite real and sort of operational. Recently, two were test-fired from a submarine. One missile hit the target half a world away, but the other missed. Why? Because of damage upon launch due to a design flaw. How long will this take to fix? Many months, followed by additional tests. Was the nuclear warhead tested? No. Will it be tested? Very unlikely. Will it be tested in the same climate and wind conditions where it would be used? Even less likely. How do we know it will detonate if it hits the target? There are computer simulations, but otherwise we don't really know. Will it detonate if it hits a wrong target? It just might. What would compel anyone to launch these missiles against such heavy risks? Only desperate circumstances.

Some weapons don't even get this far. Some become unreliable after two years in storage. Some are way too expensive to keep around (such as hundreds of Soviet-era submarines).

Having noted all that, let's discuss the DF-21. Can it cover distances greater than 1,000 miles? Sure, but that's relatively easy today. Can it move at the announced speed? Probably, although that is a very expensive proposition. Can it detonate just when it hits something, rather than be smashed by the force of impact? That's harder (at that speed), but doable. Can it hit a given target? At Mach 10, course correction is extremely difficult and must be made early enough, but you can't know how the winds, humidity, and temperature gradients will affect the full course of the missile. This is less of a problem for nuclear missiles, since their impact covers large areas, and they don't need to be very precise. But a conventional missile moving at Mach 10 may miss its target. Firing several missiles at once is an option, but this is expensive and risks interference between the missiles (note that each of them generates hot plasma around itself).

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Now, let's assume that a DF-21 can hit a stationary target the size of an aircraft carrier. Can it find the aircraft carrier among other ships in its group? This is not easy because smaller ships can increase their radar and thermal signature very quickly. When a missile launch is detected, a carrier can launch decoys around it and hide behind chemical smoke.

Even if a missile can get through those countermeasures, that's not enough because carriers constantly move (when not in port). China could track a carrier group by sending a decent-size force out to sea or from satellites, but this won't allow predicting the aircraft location accurately enough for a missile to hit it. A regular satellite cannot follow a sea target over long ranges (because it orbits much faster), and geostationary satellites have relatively poor imaging resolution. Changing the orbit is currently very expensive.

Accurate tracking requires powerful radars. China has those, but at a distance greater than 1,000 miles, their signals can easily be jammed and even spoofed by destroyers that accompany the carrier. Fake return signals can show targets at wrong places. Radars on patrol planes aren't as powerful, are too small to go over the horizon, and can also be jammed.

Even if China solves all those issues, there is no compelling way to confirm that, so any launch of an anti-ship missile at an aircraft carrier group would be a gamble. If the aircraft carrier is hit, the response from missile destroyers, cruises, and submarines will be swift. The trouble with taking that gamble is that a failed missile launch will be honored with a comparable response. In addition to new carrier groups on high alert, the U.S. will send heavy bombers and submarine-launched cruise missiles to make sure that anything resembling a DF-21 launcher is pulverized (if those launchers aren't labeled in advance or even sabotaged by special forces). Now, ask yourself who would decide to launch a DF-21 at a U.S. aircraft carrier and what would they hope to achieve, assuming everything works.

With Russia, a rule of thumb is to first look at the map (if you don't have one in your head). U.S. carriers can't enter the Black Sea due to the Montreux treaty, but the U.S. has access to the entire Black Sea from air bases in Romania and Turkey. The Baltic Sea isn't useful for carrier groups for similar reasons—Poland, Germany, Denmark, Lithuania, Latvia, and Estonia are all NATO members, and Norway is not far out either. The Russian Far East is reachable from Alaska, the Japanese islands and from U.S. carriers operating East of Japan. So, only the Arctic shore of Russia is in the game. But aircraft carriers don't operate in the presence of ice, and Russia has relatively little of value in areas that the U.S. can't reach from existing airbases or from subs.

Here's a different take on your question: If those anti-ship missiles were worth anything, would China have stayed quiet and happily watched the U.S. build “obsolete” aircraft carriers? In contrast, China knows full well that the missiles are vaporware, so it tries to get some propaganda mileage out of them. On the other hand, U.S. aircraft carriers are useful for so many purposes and in so many parts of the world (such as the Persian Gulf, Eastern and Western Mediterranean, and closer to the U.S.) that claiming their obsolescence is absurd.